# Properties

 Label 2.0.7.1-6300.2-c5 Base field $$\Q(\sqrt{-7})$$ Conductor $$(-60 a + 30)$$ Conductor norm $$6300$$ CM no Base change yes: 210.e7,1470.j7 Q-curve yes Torsion order $$16$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-7})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 2$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 2)

gp: K = nfinit(a^2 - a + 2);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);

## Weierstrass equation

$$y^2+xy=x^{3}+210x+900$$
sage: E = EllipticCurve(K, [1, 0, 0, 210, 900])

gp: E = ellinit([1, 0, 0, 210, 900],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 0, 210, 900]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-60 a + 30)$$ = $$\left(a\right) \cdot \left(-a + 1\right) \cdot \left(3\right) \cdot \left(5\right) \cdot \left(-2 a + 1\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$6300$$ = $$2^{2} \cdot 7 \cdot 9 \cdot 25$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(928972800)$$ = $$\left(a\right)^{16} \cdot \left(-a + 1\right)^{16} \cdot \left(3\right)^{4} \cdot \left(5\right)^{2} \cdot \left(-2 a + 1\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$862990463139840000$$ = $$2^{32} \cdot 7^{2} \cdot 9^{4} \cdot 25^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1023887723039}{928972800}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(6 a - 6 : -18 a - 24 : 1\right)$ Height $$0.621045073666089$$ Torsion structure: $$\Z/2\Z\times\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-12 a : 66 : 1\right)$ $\left(-\frac{45}{4} a + \frac{15}{2} : \frac{45}{8} a - \frac{15}{4} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.621045073666089$$ Period: $$0.442077482916679$$ Tamagawa product: $$4096$$  =  $$2^{4}\cdot2^{4}\cdot2\cdot2^{2}\cdot2$$ Torsion order: $$16$$ Leading coefficient: $$6.64129038694595$$ Analytic order of Ш: $$1$$ (rounded)

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$\left(-a + 1\right)$$ $$2$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$\left(-2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(3\right)$$ $$9$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(5\right)$$ $$25$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 6300.2-c consists of curves linked by isogenies of degrees dividing 32.

## Base change

This curve is the base change of elliptic curves 210.e7, 1470.j7, defined over $$\Q$$, so it is also a $$\Q$$-curve.